Exam 13: Hypothesis Testing: Comparing Two Populations

Do government employees take longer tea breaks than private-sector workers? That is the question that interested a management consultant. To examine the issue, he took a random sample of nine government employees and another random sample of nine private-sector workers and measured the amount of time (in minutes) they spent in tea breaks during the day. The results are listed below. Government employees Private sector workers 23 25 18 19 34 18 31 22 28 28 33 25 25 21 27 21 32 30 Do these data provide sufficient evidence at the 5% significance level to answer the consultant's question in the affirmative?

When testing for the difference between two population means and the population variances are unknown, a t test is used.

A psychologist has performed the following experiment. For each of 10 sets of identical twins who were born 30 years ago, she recorded their annual incomes according to which twin was born first. The results (in $000) are shown below. Twin set First born Second born 1 32 44 2 36 43 3 21 28 4 30 39 5 49 51 6 27 25 7 39 32 8 38 42 9 56 64 10 44 44 Can she infer at the 5% significance level that there is a difference in income between the twins?

Two independent samples of sizes 30 and 35 are randomly selected from two normal populations with equal variances. Which of the following is the test statistic that should be used to test the difference between the population means?

Two independent samples of sizes 30 and 40 are randomly selected from two populations to test the difference between the population means, µ?1 - µ?2, where the population variances are unknown but equal. Which of the following best describes the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ ?

In testing the hypotheses: H0: p1 - p2 = 0. HA : p1 - p2 > 0. we find the following statistics: n1 = 150, x1 = 72. n2 = 175, x2 = 70. a. What is the p-value of the test? b. Briefly explain how to use the p-value to test the hypotheses, at a 1% level of significance.

The test statistic to test the difference between two population proportions is the Z test statistic, which requires that the sample sizes are each sufficiently large.

In testing the difference between two population means, using two independent samples, the sampling distribution of the sample mean difference $\bar { X } _ { 1 } - \bar { X } _ { 2 }$ is normal if the:

A marketing consultant is studying the perceptions of married couples concerning their weekly food expenditures. He believes that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of 10 married couples and asks each spouse to estimate the family food expenditure (in dollars) during the previous week. The data are shown below. Couple Husband Wife 1 380 270 2 280 300 3 215 185 4 350 320 5 210 180 6 410 390 7 250 250 8 360 320 9 180 170 10 400 330 Can the consultant conclude at the 5% significance level that the husband's estimate is higher than the wife's estimate?

The managing director of a breakfast cereal manufacturer claims that families in which both spouses work are much more likely to be consumers of his product than those with only one working spouse. To prove his point, he commissions a survey of 300 families in which both spouses work and 300 families with only one working spouse. Each family is asked whether the company's cereal is eaten for breakfast. The results are shown below. Two spouses working One spouse working Eat cereal 114 87 Do not eat cereal 186 213 Use the p-value method to test whether the proportion of families with two working spouses who eat the cereal is at least 5% larger than the proportion of families with one working spouse who eats the cereal, at the 1% level of significance.

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population proportions. The two sample proportions are p1 = 0.25 and p2 = 0.20 and the standard error of the sampling distribution of $\hat { p } _ { 1 } - \hat { p } _ { 2 }$ is 0.04. The calculated value of the test statistic will be:

A course coordinator at a university wants to investigate if there is a significant difference in the average final mark of students taking the same university subject in semester 1 or semester 2. A random sample of 30 students is taken from semester 1, with the average final mark is found to be 60 and the standard deviation is 5. A random sample of 50 students is taken from semester 2, with the average final mark is 57 and the standard deviation is 4. Assuming that the population variances are equal, is there significant evidence that the population average final mark in this course differs between semester 1 and semester 2. Test at the 5% level of significance.

An industrial statistician wants to determine whether efforts to promote safety have been successful. By checking the records of 250 employees, he finds that 30 of them have suffered either minor or major injuries that year. A random sample of 400 employees taken in the previous year revealed that 80 had suffered some form of injury. a. Can the statistician infer at the 5% significance level that efforts to promote safety have been successful? b. What is the p-value of the test?

In testing the hypotheses H0: p1 - p2 = 0 HA: p1 - p2 ≠ 0, we find the following statistics: n1 = 400, x1 = 105. n2 = 500, x2 = 140. What conclusion can we draw at the 10% significance level?

When the necessary conditions are met, a two-tail test is being conducted to test the difference between two population means, but your statistical software provides only a one-tail area of 0.028 as part of its output. The p-value for this test will be:

In testing the hypotheses: H0: p1 - p2 = 0 HA : p1 - p2 ≠ 0, we find the following statistics: n1 = 150, x1 = 72. n2 = 175, x2 = 70. What conclusion can we draw at the 5% significance level?

In testing the null hypothesis H0: p1 - p2 = 0, if H0 is true, the test could lead to:

A politician has commissioned a survey of blue-collar and white-collar employees in her electorate. The survey reveals that 286 out of 542 blue-collar workers intend to vote for her in the next election, whereas 428 out of 955 white-collar workers intend to vote for her. a. Can she infer at the 5% level of significance that the level of support differs between the two groups of workers? b. What is the p-value of the test? Explain how to use it to test the hypotheses.

The degrees of freedom for a t test of the difference of population means from two independent samples are n1 + n2 - 2.

The number of degrees of freedom associated with the t-test, when the data are gathered from a matched pairs experiment with 40 pairs, is: